Statistics Introduction 2. The word Probability derives from the Latin probabilitas, which can also mean probity, a measure of the authority of a witness.

Statistics Introduction 2

• The word Probability derives from the Latin probabilitas, which can also mean probity, a measure of the authority of a witness in a legal case in Europe, and often correlated with the witness's nobility.

• In a sense, this differs much from the modern meaning of probability, which, in contrast, is a measure of the weight of empirical evidence, and is arrived at from inductive reasoning and statistical inference.

True Probability vs. experimental probability vs. theoretical probability

• We tend to give answers to probability questions based on ‘theoretical’ probability but all three types are usually involved in a problem.

TRUE probability

• True probability involves an exact understanding of all the factors involved that lead to a certain outcome.

• It is a deterministic model.

Deterministic model

• Mathematical model in which outcomes are precisely determined through known relationships among states and events, without any room for random variation.

Deterministic Model

• In such models, a given input will always produce the same output, such as in a known chemical reaction.

• A deterministic system is a system in which no randomness is involved in the development of future states of the system.

The important idea

• A deterministic model will thus always produce the same output from a given starting condition or initial state.

Deterministic Model

• With a deterministic model, the assumptions and equations you select "determine" the results. The only way the outputs change is if you change an assumption (or an equation).

Experimental Probability

• Experimental probability, is the ratio of the number of outcomes in which a specified event occurs to the total number of trials, not in a theoretical sample space but in an actual experiment.

• In a more general sense, experimental probability estimates probabilities from experience and observation.

Advantage of experimental probability

• An advantage of estimating probabilities using experimental probabilities is that this procedure is relatively free of assumptions.

• For example, consider estimating the probability among a population of men that they satisfy two conditions:

• 1. that they are over 2 metres in height• 2. that they prefer strawberry jam to raspberry

jam.• A direct estimate could be found by counting the

number of men who satisfy both conditions to give the experimental probability of the combined condition.

• An alternative estimate could be found by multiplying the proportion of men who are over 2 metres in height with the proportion of men who prefer strawberry jam to raspberry jam, but this estimate relies on the assumption that the two conditions are statistically independent.

Disadvantage

• A disadvantage in using experimental probabilities arises in estimating probabilities which are either very close to zero, or very close to one.

• In these cases very large sample sizes would be needed in order to estimate such probabilities to a good standard of relative accuracy.

Theoretical Probability

• Theoretical probability is the probability that a certain outcome will occur, as determined through reasoning or calculation.

Example:

• Given a die which is a regular octahedron of uniform density, and given that one and only one of its faces is painted black, then if the die is cast, the theoretical probability that the outcome will be the black face is 1/8.

Experimental/Theoretical probability

• In many cases, the quality of a scientific field depends on how well the mathematical models developed on the theoretical side agree with results of repeatable experiments.

• Lack of agreement between theoretical mathematical models and experimental measurements often leads to important advances as better theories are developed.

Summary

Type Model Likely occurrence of future event

True Probability

Deterministic Usually unknown

Experimental probability

Probabilistic Applying past to the future

Theoretical probability

Probabilistic Based on a model

Babies

• Worldwide, we rely on the assumption that a newborn baby is equally likely to be a boy or a girl.

• Is this in fact true?

True Probability

• This would involve an exact understanding of all the factors involved that lead to a certain outcome.

• The process by which gender is determined is largely unknown because many other factors other than X and Y chromosomes may be involved.

• Is gender influenced more by the father or by the mother?

• Does gender depend on the age of the mother and/or of the father?

• Is gender related to the parents’ occupations?

• Is there a hereditary factor involved?

• Are there only two possible outcomes?

• Are there global or economic influences?

• ???????

True Probability

• If we knew all the determining factors, we would be able to predict the gender of the baby.

http://www.webmd.com/baby/features/predicting-your-babys-sex

Experimental probability

• Experimental probability relies on looking at long-run relative frequencies from the past and projecting these patterns into the future.

http://en.wikipedia.org/wiki/List_of_countries_by_sex_ratio

Experimental factors that may influence the experimental probability

• One gender may be more likely to survive from conception to birth than the other.

• Live-birth ratios are not necessarily the same as gender ratios at conception.

• Some cultures practise abortion or infanticide because one gender is seen to be more desirable than the other.

• Some providers of artificial insemination or in vitro fertilisation (test-tube babies) offer gender selection

Theoretical probability

• This involves a mathematical model to explain the distributions of various outcomes.

• We generally assume that the 50:50 model applies and use the probabilities to make predictions.

Comments

• The 50:50 model is probably close enough for most people to believe this is true.

• There is an unproven belief that, because there appears to be two outcomes and the process is apparently random, then the probabilities are equal. However, random events do not have to be equally likely.

http://www.swissinfo.ch/eng/swiss_news/Third_gender_fights_for_recognition.html?cid=34791620

Deterministic and probabilistic models

• A deterministic model does not include elements of randomness. Every time you run the model with the same initial conditions you will get the same results.

• A probabilistic model does include elements of randomness. Every time you run the model, you are likely to get different results, even with the same initial conditions.

Randomness

• http://www.youtube.com/watch?v=Lf4ZmWc_jmA

• Randomly put one cross on your paper.

• Where are most of the crosses?

What does randomness look like?

file://localhost/Users/marionsteel/Desktop/workshops/probability workshop/random scatter.xls Or is this random scatter?Is this random scatter?

Benford’s law

• http://www.nctm.org/resources/content.aspx?id=7636

Benford’s Law

• This result has been found to apply to a wide variety of data sets, including electricity bills, street addresses, stock prices, population numbers, death rates, lengths of rivers, physical and mathematical constants, and processes described by power laws (which are very common in nature). It tends to be most accurate when values are distributed across multiple orders of magnitude.

TED Talk

• http://www.ted.com/talks/peter_donnelly_shows_how_stats_fool_juries.html

Is there such a thing as a biased coin?